This work describes and presents the properties of a proposed tuning method, which is compared with the well known pythagorean tuning method. These two methods are shown to be derived with a few algebraic rules. The two algorithms are compared in terms of their simplicity, correspondance to the consonance of intervals, and errors with the well-established twelve-tone equal temperament tuning method. The mathematics behind the proposed inverse fraction rule is explained.
For about a month, the original idea of the tuning system was in the back of my mind. Then, I thought of the inverse fraction rule and immediately spent a good part of a morning quickly writing the vast majority of this notebook.
In Western music, the most common tuning system since the 18th century has been twelve-tone equal temperament, which divides the octave into 12 parts, all of which are equal on a logarithmic scale. The frequency interval between every pair of adjacent notes has the same ratio equal to the 12th root of 2: $\large \sqrt[12]{2}$ ≈ 1.05946. Intervals are then described with $\large 2^\frac{n}{12}$ where n represents the number of semitones, also called halfsteps.
In the twelve-tone equal temperament, the inverse of an interval at n semitones is the interval at 12 - n semitones. For example, the inverse of a perfect fifth (7 semitones) is a perfect fourth (12 - 7 = 5 semitones).
Begin with the fraction $\large \frac{2}{1}$
For example: We start with $\large \frac{2}{1}$, a perfect octave. Following the 1st rule: $\large \frac{2}{1}$ -> $\large \frac{2}{1 \times 2}$ -> $\large \frac{2}{2} = \frac{1}{1}$, a perfect unison (the inverse of a perfect octave). Then, following the 2nd rule: $\large \frac{2 + 1}{1 + 1} = \frac{3}{2}$, a perfect fith. Etc.
Begin with the calculation $\large (\frac{2}{3})^6 (2)^4$, which is $\large \frac{1024}{729}$
For example: We start with $\large (\frac{2}{3})^6 (2)^4 = \frac{1024}{729}$, a diminished fifth. Following the 1st rule: $\large (\frac{2}{3})^5 (2)^3 = \frac{256}{243}$, a minor second. Then, following the 2nd rule: $\large (\frac{2}{3})^4 (2)^3 = \frac{128}{81}$, a minor sixth. Etc.
Some of the proposed fractions are reducible wheras all the Pythagorean fractions are irreducible.
The inverse fraction rule that is fundamental to my algorithm suggests the following:
Given two positive integers $a$ and $b$, and an integer $i$ from 0 through 12, inclusive.
If:
$$\large \frac{a}{b} \approx 2^\frac{i}{12}$$Then:
$$\large \frac{2b}{a} \approx 2^\frac{12 - i}{12}$$Define $\epsilon$ as the initial error, and let $\gamma = \epsilon + 1$. Then:
$$\large \frac{a}{b} = \gamma k^\frac{i}{n}$$Where in the case of the tuning algorithm, $n = 12$ and $k = 2$. Rearranging algebraiclly we get:
$$\large \frac{kb}{a} = \frac{1}{\gamma} k^{\frac{n-i}{n}}$$As $\epsilon \rightarrow 0$, we have that $\gamma \rightarrow 1$. Therefore:
$$\large \frac{kb}{a} \rightarrow k^{\frac{n-i}{n}}$$import tuning
tuning.plot_errors()
tuning.plot_abs_errors()
tuning.show_algorithm()